Classical Segal-Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $\mathbb{R}^n$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $\mathbb{R}^{n+1}$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of $\mathbb{R}^n$.
We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.
We study the Segal-Bargmann transform on a motion group Rn n K; where K is a compact subgroup of SO(n): A characterization of the Poisson integrals associated to the Laplacian on Rn n K is given. We also establish a Paley-Wiener type theorem using the complexified representations.
In this note we investigate the image of Sobolev spaces of fractional order on a compact Lie group $ K $ under the Segal-Bargmann transform. We show that the image can be characterised in terms of certain weighted Bergman spaces of holomorphic functions on the complexification extending a theorem of Hall and Lewkeeratiyutkul. We also treat the heat kernel transform associated to the Hermite operator.
We study the Segal-Bargmann transform on $M(2).$ The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are studied. Using a Gutzmer type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the inverse Fourier transform
We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex $n$-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sjöstrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sjöstrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform.
We study the Segal-Bargmann transform on the Heisenberg motion groups $\mathbb{H}^n \ltimes K,$ where $\mathbb{H}^n$ is the Heisenberg group and $K$ is a compact subgroup of $U(n)$ such that $(K,\mathbb{H}^n)$ is a Gelfand pair. The Poisson integrals associated to the Laplacian for the Heisenberg motion group are also characterized using Gutzmer's formulae. Explicitly realizing certain unitary irreducible representations of $\mathbb{H}^n \ltimes K,$ we prove the Plancherel theorem. A Paley-Wiener type theorem is proved using complexified representations.
For $E$ a Hilbert space, let $\mathcal{H}(E)$ denote the Segal-Bargmann space (also known as the Fock space) over $E$, which is a reproducing kernel Hilbert space with kernel $K(x,y)=\exp(< x,y>)$ for $x,y$ in $E$. If $ϕ$ is a mapping on $E$, the composition operator $C_ϕ$ is defined by $C_ϕh = h\circϕ$ for $h\in \mathcal{H}(E)$ for which $h\circϕ$ also belongs to $\mathcal{H}(E)$. We determine necessary and sufficient conditions for the boundedness and compactness of $C_ϕ$. Our results generalize results obtained earlier by Carswell, MacCluer and Schuster for finite dimensional spaces $E$.
We study the complex-time Segal-Bargmann transform $\mathbf{B}_{s,τ}^{K_N}$ on a compact type Lie group $K_N$, where $K_N$ is one of the following classical matrix Lie groups: the special orthogonal group $\mathrm{SO}(N,\mathbb{R})$, the special unitary group $\mathrm{SU}(N)$, or the compact symplectic group $\mathrm{Sp}(N)$. Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group $\mathrm{U}(N)$. We provide an effective method of computing the action of the Segal-Bargmann transform on \emph{trace polynomials}, which comprise …
We study the (two-parameter) Segal--Bargmann transform $\mathbf{B}_{s,t}^N$ on the unitary group $\mathbb{U}_N$, for large $N$. Acting on matrix valued functions that are equivariant under the adjoint action of the group, the transform has a meaningful limit $\mathscr{G}_{s,t}$ as $N\to\infty$, which can be identified as an operator on the space of complex Laurent polynomials. We introduce the space of {\em trace polynomials}, and use it to give effective computational methods to determine the action of the heat operator, and thus the Segal--Bargmann transform. We prove several concentration o…
We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of Śniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.
Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the large-$N$ limit of the two-parameter Segal-Bargmann transform on the unitary group $\mathbb{U}(N)$ and explore its limiting behavior. We also extend the notion of circular systems in order to define a two-parameter free Segal-Bargmann transform and prove a version of Biane-Gross-Malliavin Theorem of the two-parameter free unitary Segal-Bargmann transform.
The Segal-Bargmann transform is a Lie algebra and Hilbert space isomorphism between real and complex representations of the oscillator algebra. The Segal-Bargmann transform is useful in time-frequency analysis as it is closely related to the short-time Fourier transform. The Segal-Bargmann space provides a useful example of a reproducing kernel Hilbert space. Coupled supersymmetries (coupled SUSYs) are generalizations of the quantum harmonic oscillator that have a built-in supersymmetric nature and enjoy similar properties to the quantum harmonic oscillator. In this paper, we will develop Sega…
We study dilated holomorphic $L^p$ space of Gaussian measures over $\mathbb{C}^n$, denoted $\mathcal{H}_{p,α}^n$ with variance scaling parameter $α>0$. The duality relations $(\mathcal{H}_{p,α}^n)^\ast \cong \mathcal{H}_{p',α}$ hold with $\frac{1}{p}+\frac{1}{p'}=1$, but not isometrically. We identify the sharp lower constant comparing the norms on $\mathcal{H}_{p',α}$ and $(\mathcal{H}_{p,α}^n)^\ast$, and provide upper and lower bounds on the sharp upper constant. We prove several suggestive partial results on the sharpness of the upper constant. One of these partial results leads to a sharp …
For $α>0$, the Bargmann projection $P_α$ is the orthogonal projection from $L^2(γ_α)$ onto the holomorphic subspace $L^2_{hol}(γ_α)$, where $γ_α$ is the standard Gaussian probability measure on $\C^n$ with variance $(2α)^{-n}$. The space $L^2_{hol}(γ_α)$ is classically known as the Segal-Bargmann space. We show that $P_α$ extends to a bounded operator on $L^p(γ_{αp/2})$, and calculate the exact norm of this scaled $L^p$ Bargmann projection. We use this to show that the dual space of the $L^p$-Segal-Bargmann space $L^p_{hol}(γ_{αp/2})$ is an $L^{p'}$ Segal-Bargmann space, but with the Gaussian …
The authors provide a correct proof of Theorem 1.2 and correct the statement of Theorem 2.6 in their paper, which appeared in Trans. Amer. Math. Soc. 375 (2022), 3733-3753.
We study systems of holomorphic Hermite functions in the Segal-Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose anihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient con…
We study the hypercyclic, supercyclic and cyclic properties of composition operator $C_ϕ$ on the Segal-Bargmann space $\mathscr{H}(\mathscr{E})$, where $ϕ(z)=Az+b$, $A\in \mathcal{B}(\mathscr{E})$, $b\in \mathscr{E}$ with $\left\|A\right\|\leq 1$ and $A^*b\in (I-A^*A)^{\frac{1}{2}}$. In this connection we also give a characterization of the symbols $ϕ$ which induce the bounded composition operator $C_ϕ$ on $\mathscr{H}(\mathscr{E})$ and show that the properties of $ϕ$ influence the cyclic behaviour of $C_ϕ$.
We introduce a new form of the Segal--Bargmann transform for a Lie group $K$ of compact type. We show that the heat kernel $(ρ_{t}(x))_{t>0,x\in K}$ has a space-time analytic continuation to a holomorphic function \[ (ρ_{\mathbb{C}}(τ,z))_{\mathrm{Re}\,τ>0,z\in K_{\mathbb{C}}} \] where $K_{\mathbb{C}}$ is the complexification of $K$. The new transform is defined by the integral \[ (B_τf)(z)=\int_{K}ρ_{\mathbb{C}}(τ,zk^{-1})f(k)\,dk,\quad z\in K_{\mathbb{C}}. \] If $s>0$ and $τ\in\mathbb{D}(s,s)$ (the disk of radius $s$ centered at $s$), this integral defines a holomorphic function on $K_{\math…
We study the large-$N$ limit of the Segal--Bargmann transform on $S^{N-1}(\sqrt N)$, the $(N-1)$-dimensional sphere of radius $\sqrt N$, as a unitary map from the space of square-integrable functions with respect to the normalized spherical measure onto the space of holomorphic square-integrable functions with respect to a certain measure on the quadric. In particular, we give an explicit formulation and describe the geometric models for the limit of the domain, the limit of the range, and the limit of the transform when $N$ tends to infinity. We show that the limiting transform is still a uni…
Let $\mathcal{D}=G/K$ be a complex bounded symmetric domain of tube type in a complex Jordan algebra $V$ and let $\mathcal{D}_{\mathbb{R}}=H/L\subset \mathcal{D}$ be its real form in a formally real Euclidean Jordan algebra $J\subset V$. We consider representations of $H$ that are gotten by the generalized Segal-Bargmann transform from a unitary $G$-space of holomorphic functions on $\mathcal{D}$ to an $L^2$-space on $\mathcal{D_{\mathbf{R}}}$. We prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the Laplace transform. We find the extens…
The paper considers bounded linear radial operators on the polyanalytic Fock spaces $\mathcal{F}_n$ and on the true-polyanalytic Fock spaces $\mathcal{F}_{(n)}$. The orthonormal basis of normalized complex Hermite polynomials plays a crucial role in this study; it can be obtained by the orthogonalization of monomials in $z$ and $\overline{z}$. First, using this basis, we decompose the von Neumann algebra of radial operators, acting in $\mathcal{F}_n$, into the direct sum of some matrix algebras, i.e. radial operators are represented as matrix sequences. Secondly, we prove that the radial opera…
We introduce an extended class of cross-Toeplitz operators which act between Fock--Segal--Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-varian…
We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schrödinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary time and the evolution of polarized sections has interesting features. On the time intervals for which the polarization is real or Kähler, the half--form corrected time evolution of polarized sections is given by unitary operators which turn out to be equivalent to the classical Segal-Bargmann transforms (which are usually associated to the quadratic elliptic Ham…
We describe a construction of wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example we consider operator valued Segal-Bargmann type spaces and the Weyl functional calculus. Keywords: Wavelets, coherent states, Banach spaces, group representations, covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann spaces, Weyl functional calculus (quantization), second quantization, bosonic field.
We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal-Bargmann space of holomorphic functions. Connections with quantum mechanics are described. Keywords: Segal-Bargmann space, Heisenberg group, coherent states, wavelet transform, reproducing kernel, nilpotent Lie group, monogenic functions, Dirac operator, Clifford algebra, (second) quantization, quantum field theory.
We introduce a one-parameter family of transforms, $U^t_{(m)}$, $t>0$, from the Hilbert space of Clifford algebra valued square integrable functions on the $m$--dimensional sphere, $L^2(S^{m},dσ_{m})\otimes \mathbb{C}_{m+1}$, to the Hilbert spaces, ${\mathcal M}L^2(\mathbb{R}^{m+1} \setminus \{0\},dμ_t)$, of monogenic functions on $\mathbb{R}^{m+1}\setminus \{0\}$ which are square integrable with respect to appropriate measures, $dμ_t$. We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal-Bargman coherent state transform, $U_{(1)} : L^2(…
This is (raw) lecture notes of the course read on 6th European intensive course on Complex Analysis (Coimbra, Portugal) in 2000. Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained. The framework is based on wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Reduced wavelet transform allows naturally describe in abstract term main objects of an analytical function theory: the Cauchy integral formula, the Hardy and Bergman spaces,…
We look at Toeplitz operators $T_ν$ on the Fock Space (also known as the Segal-Bargmann space) which have a positive Borel measure $ν$ as a symbol. We characterize when $\left(T_ν\right)^s$ for $0<s\leq 1$ is in the symmetrically normed ideal associated with any given symmetric norming function.
In 1990 van Eijnghoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand-Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present p…
Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Car…
The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we …
We study a transform, inspired by coherent state transforms, from the Hilbert space of Clifford algebra valued square integrable functions $L^2({\mathbb R}^m,dx)\otimes {\mathbb C}_{m}$ to a Hilbert space of solutions of the Weyl equation on ${\mathbb R}^{m+1}= {\mathbb R} \times {\mathbb R}^m$, namely to the Hilbert space ${\mathcal M}L^2({\mathbb R}^{m+1},dμ)$ of ${\mathbb C}_m$-valued monogenic functions on ${\mathbb R}^{m+1}$ which are $L^2$ with respect to an appropriate measure $dμ$. We prove that this transform is a unitary isomorphism of Hilbert spaces and that it is therefore an analo…
The Bargmann-Fock-Segal space plays an important role in mathematical physics, and has been extended into a number of directions. In the present paper we imbed this space into a Gelfand triple. The spaces forming the Fréchet part (i.e. the space of test functions) of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable, using the Stirling numbers of the second kind. The dual of the space of test functions has a topological algebra structure, of the kind introduced and studied by the first named author and G. Salomon.
We continue the construction of the $:ϕ^4_4:$ quantum field theory. In this paper we consider the Wick kernel of the interacting quantum field. Using the complex structure and the Fock-Bargmann-Berezin-Segal integral representation we prove that this kernel defines a unique operator--valued generalized function on the space $\Sc^α(\R^4)$ for any $α<6/5,$ i.e. the constructed quantum field is the generalized operator-valued function of localizable Jaffe class. The same assertion is valid for the outgoing quantum field. These assertions about the quantum field allow to construct the Wightman fun…
This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is systematically illustrated by a representative collection of examples. Keywords: Lie groups and algebras, convolution operator, relative convolution, representation theory, pseudo-differential operators, PDO, singular integral operator, SIO, the Heisenberg group, SL(2,R), induced representation, Fock-Segal-Bargmann (FSB) representation, Shrodinger representation, covar…
